Provably Good Sampling and Meshing of Surfaces
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Transcript of Provably Good Sampling and Meshing of Surfaces
Provably Good Sampling and Meshing of Surfaces
Jean-Daniel Boissonnat, Steve Oudot
Provably Good Sampling and Meshing of Surfaces
Not a smooth surface
Jean-Daniel Boissonnat, Steve Oudot
Smooth surface
Provably Good Sampling and Meshing of Surfaces
Jean-Daniel Boissonnat, Steve Oudot
Provably Good Sampling and Meshing of Surfaces
Well distributed sample points
Smooth surface
Jean-Daniel Boissonnat, Steve Oudot
Provably Good Sampling and Meshing of Surfaces
Good triangula9on
Jean-Daniel Boissonnat, Steve Oudot
Well distributed sample points
Smooth surface
Provably Good Sampling and Meshing of Surfaces
25 All angles are greater than 25 degrees
Jean-Daniel Boissonnat, Steve Oudot
Good triangula9on:
Well distributed sample points
Smooth surface
Provably Good Sampling and Meshing of Surfaces
25
All triangles are equilateral
Jean-Daniel Boissonnat, Steve Oudot
Good triangula9on:
Well distributed sample points
Smooth surface
All angles are greater than 25 degrees
Provably Good Sampling and Meshing of Surfaces
25
The best approximates
Jean-Daniel Boissonnat, Steve Oudot
All triangles are equilateral
All angles are greater than 25 degrees
Smooth surface
Good triangula9on:
Well distributed sample points
Provably Good Sampling and Meshing of Surfaces
25
The best approximates
All triangles are equilateral
All angles are greater than 25 degrees
Smooth surface
Jean-‐Daniel Boissonnat, Steve Oudot
Good triangula9on:
Well distributed sample points
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1. Take a smooth surface Compact, orientable, at least C2 – con9nuous closed surface.
Completely suitable Not completely suitable
2. Sample this surface • Medial axis of the surface
2d medial axis 3d medial axis
M S
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2. Sample this surface • Distance to the medial axis that is
2d medial axis 3d medial axis
dM
dM
dMM
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2. Sample this surface • Minimum distance to the medial axis that is
dMinf = inf dM (x), x ∈ S{ }
dMinfM
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2. Sample this surface • Some user-‐defined func9on σ : S→ R
Ø Posi9ve that is
Ø 1-‐Lipschitz that is σ (x)−σ (y) ≤ x − y
σ > 0
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2. Sample this surface • Ball of center and radius that is B(c, r)cB r
B(c, r)
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2. Sample this surface • Ball of center and radius that is B(c, r)cB r
B(c, r)
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2. Sample this surface • Construc9on of the ini9al point sample E
Pick up at least one point x on each connected component of S and insert it in !E
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2. Sample this surface • Construc9on of the ini9al point sample E
Consider a ball Bx centered at x of radius less
min 16dist(x, !E \ {x}),dM (x),
16σ (x)
"#$
%&'
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2. Sample this surface • Construc9on of the ini9al point sample E
Repeatedly shoot rays inside Bx and pick up three points (ux, vx, wx) of S Bx
Insert (ux, vx, wx) in E
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2. Sample this surface • Construc9on of the ini9al point sample E
Connec9ng these points we get a persistent facet
All persistent facets are Delaunay facets restricted to S
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2. Sample this surface • Construc9on of the ini9al point sample E
All persistent facets remain restricted Delaunay facets throughout the course of algorithm
Presented by Anisimov Dmitry
2. Sample this surface • Construc9on of the ini9al point sample E
All persistent facets remain restricted Delaunay facets throughout the course of algorithm
Presented by Anisimov Dmitry
2. Sample this surface • Construc9on of the ini9al point sample E
All persistent facets remain restricted Delaunay facets throughout the course of algorithm
Presented by Anisimov Dmitry
2. Sample this surface • Construc9on of the ini9al point sample E
All persistent facets remain restricted Delaunay facets throughout the course of algorithm
Presented by Anisimov Dmitry
2. Sample this surface • Construc9on of the ini9al point sample E
All persistent facets remain restricted Delaunay facets throughout the course of algorithm
Presented by Anisimov Dmitry
2. Sample this surface • Construc9on of the ini9al point sample E
All persistent facets remain restricted Delaunay facets throughout the course of algorithm
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3. Triangulate this surface • Compute the 3-‐dimensional Delaunay triangula9on of E
Del(E)
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3. Triangulate this surface • Compute the set of all edges of the Voronoi diagram of E
V(E)
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3. Triangulate this surface • Compute Delaunay triangula9on of E restricted to S
DelS(E)
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3. Triangulate this surface • Compute Delaunay triangula9on of E restricted to S
DelS(E)
Not constrained
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3. Triangulate this surface • Surface Delaunay ball BD of restricted Delaunay facet f
DelS(E)
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3. Triangulate this surface • Surface Delaunay ball BD of restricted Delaunay facet f
DelS(E)
Any ball centered at some point of where f* is Voronoi edge dual to f
S f *
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3. Triangulate this surface • Bad surface Delaunay ball BD which is stored in L
DelS(E)
c
It is ball B(c, r) such that r > σ(c)
r
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3. Triangulate this surface • Surface Delaunay patch
DelS(E)
The intersec9on of a surface Delaunay ball with S
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3. Triangulate this surface • Loose ε-sample
dM c
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3. Triangulate this surface • E is a loose ε-‐sample of S if:
dM c
1. ∀c ∈ SV (E),E B(c,εdM (c)) ≠∅2. DelS(E) has ver9ces on all the connected components of S
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3. Triangulate this surface • DelS(E) has ver9ces on all the connected components of S
V(E)
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3. Triangulate this surface • Algorithm While L is not empty
• Take an element B(c,r) from L • Insert c into E and update Del(E) • Update DelS(E) by tes9ng all the Voronoi edges that have changed or appeared:
Ø Delete from DelS(E) the Delaunay facets whose dual Voronoi edges no longer intersect S Ø Add to DelS(E) the new Delaunay facets whose dual Voronoi edges intersect S
• Update L by Ø Dele9ng all the elements of L which are no longer bad surface Delaunay balls Ø Adding all the new surface Delaunay balls that are bad
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3. Triangulate this surface • Termina9on and output of the Algorithm
Ø The Algorithm terminates
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3. Triangulate this surface • Termina9on and output of the Algorithm
Ø The Algorithm outputs E and DelS(E)
E is a loose ε-‐sample of S
DelS(E) is homeomorphic to the input surface S and approximates it in terms of its Hausdorff distance, normals, curvature, and area.
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3. Triangulate this surface • Output of the Algorithm
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Magic Epsilon • To find ε you must solve this simple inequality:
2ε1−8ε
+ arcsin ε1−ε
≥π4
• Or just take this:
ε = 0.091
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Applica9ons
Smooth surface
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Applica9ons
Not smooth surface
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Applica9ons
Bad triangula9on Good triangula9on
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References J.-‐D. Boissonnat and S. Oudot. “Provably Good Sampling and Meshing of Surfaces.” Graphical Models 67 (2005), 405-‐51.
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References M. Botsch, L. Kobbelt, M. Pauly, P. Alliez, and B. Levy. “Polygon Mesh Processing.” Chapter 6, Sec9on 6.5.1 (2010), 92-‐96.
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What steps?
Did I forget something?
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